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$$\text{The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than 500 people. When the freshmen are told to line up in columns of 23, 22 people are in the last column. When the freshmen are told to line up in columns of 21, 14 people are in the last column. How many people are in the incoming freshman class? }$$

Aug 10, 2019

#1
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This short computer code gives the solutions as follows:

a=1; b=2;c=23*a + 22; d=21*b+14;if(c==d, goto5, goto6);printc, a, b; a++;if(a<100, goto2, 0);a=1;b++;if(b<100, goto2, discard=0;

23*a + 22 = 21*b +14

a = 17 and b = 19  for a total = 413 - Number of students in freshman's class.

a = 38 and b = 42 for a total = 896 - which is above the stated number of students.

Aug 10, 2019
#2
+23293
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The American Mathematics College is holding its orientation for incoming freshmen.
The incoming freshman class contains fewer than $$500$$ people.
When the freshmen are told to line up in columns of $$23$$, $$22$$ people are in the last column.
When the freshmen are told to line up in columns of $$21$$, $$14$$ people are in the last column.
How many people are in the incoming freshman class?

$$\begin{array}{|rclrclr|} \hline x &\equiv& 22 \pmod{23} \\ \text{or } \quad x &=& 22+23n \\ x &\equiv& 14 \pmod{21} \\ \text{or}\quad \mathbf{ x } &=& \mathbf{14+21m} \\ \hline x = 14+21m &=& 22+23n \\ 21m &=& 23n + 8 \\ m &=& \dfrac{23n + 8}{21} \\ m &=& \dfrac{21n +2n + 8}{21} \\ m &=& n+ \underbrace{\dfrac{2n + 8}{21}}_{=a} \\ \mathbf{ m } &=& \mathbf{n+a} & a &=& \dfrac{2n + 8}{21} \\ & & & 21a &=& 2n + 8 \\ &&& 2n &=& 21a-8 \\ &&& n &=& \dfrac{21a-8}{2} \\ &&& n &=& \dfrac{20a+a-8}{2} \\ &&& n &=& 10a+\underbrace{\dfrac{a-8}{2}}_{=b} \\ &&& \mathbf{ n } &=& \mathbf{ 10a+b} & b=\dfrac{a-8}{2} \\ &&& & & & 2b= a-8 \\ &&& & & & \mathbf {a= 2b+8} \\ &&& n &=& 10(2b+8)+b \\ &&& \mathbf{n} &=& \mathbf{21b+80} \\ m &=& 21b+80+2b+8 \\ \mathbf{m} &=& \mathbf{23b+88} \\ \hline x &=& 14+21(23b+88) \\ x &=& 1862+483b \quad | \quad 1863 \mod{483} \equiv 413 \\ \mathbf{x} &=& \mathbf{413+483b} \\ \hline \end{array}$$

In the incoming freshman class are 413 people

Aug 11, 2019